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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 439530.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
439530.i1 | 439530i7 | \([1, 1, 0, -24439952583, 1470600247336053]\) | \(13722604572968640187892492722921/36939806611960382108160\) | \(4345931308090526994642915840\) | \([2]\) | \(955514880\) | \(4.5376\) | \(\Gamma_0(N)\)-optimal* |
439530.i2 | 439530i8 | \([1, 1, 0, -4343210183, -81360531892107]\) | \(77013704252633562960444236521/20262661472595628847255040\) | \(2383881859589403138250708200960\) | \([2]\) | \(955514880\) | \(4.5376\) | |
439530.i3 | 439530i5 | \([1, 1, 0, -4019439008, -98085244614552]\) | \(61042428203425827148268361721/2287149206968899000\) | \(269080817050683998451000\) | \([2]\) | \(318504960\) | \(3.9883\) | |
439530.i4 | 439530i6 | \([1, 1, 0, -1546525383, 22375778720373]\) | \(3477015524751011858387583721/173605868128473455001600\) | \(20424556779446773507483238400\) | \([2, 2]\) | \(477757440\) | \(4.1910\) | \(\Gamma_0(N)\)-optimal* |
439530.i5 | 439530i4 | \([1, 1, 0, -403729008, 537193271448]\) | \(61859347930211625693801721/34737934177406743101000\) | \(4086883218037725919089549000\) | \([2]\) | \(318504960\) | \(3.9883\) | \(\Gamma_0(N)\)-optimal* |
439530.i6 | 439530i2 | \([1, 1, 0, -251584008, -1527931671552]\) | \(14968716721822395621081721/91209357028881000000\) | \(10730689645090820769000000\) | \([2, 2]\) | \(159252480\) | \(3.6417\) | \(\Gamma_0(N)\)-optimal* |
439530.i7 | 439530i1 | \([1, 1, 0, -6584008, -51414671552]\) | \(-268291321601301081721/9550359000000000000\) | \(-1123590185991000000000000\) | \([2]\) | \(79626240\) | \(3.2951\) | \(\Gamma_0(N)\)-optimal* |
439530.i8 | 439530i3 | \([1, 1, 0, 59106617, 1368653011573]\) | \(194108149567956675968279/6990401110687088640000\) | \(-822413700271225291407360000\) | \([2]\) | \(238878720\) | \(3.8444\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 439530.i have rank \(0\).
Complex multiplication
The elliptic curves in class 439530.i do not have complex multiplication.Modular form 439530.2.a.i
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.